During the past decades, some progress have been made in the subject.
Under the assumption of zero virtual masses, the Feynman integral of one-loop triangle
diagram is given as linear combination of the fourth kind of Appell function [1]} whose
arguments are the dimensionless ratios among the external momentum squared, and is simplified
further as the linear combination of the Gauss function \(_2F_1\)  through the quadratic
transformation [2]} in the literature [3]}.
Taking some special assumptions on parameter space, the author of Ref. [4]}
obtains the multiple hypergeometric expressions of the scalar integral \(C_{_0}\)  through the
Mellin-Barnes representations. An algorithm to evaluate the scalar integrals of
one-loop vertex-type Feynman diagrams is presented in Ref [5]},
and a geometrical interpretation of the analytic expressions of the scalar integrals
of one-loop \(N-\) point Feynman diagrams is also given in Ref [6]}.
Certainly the expressions of \(C_{_0}\)  function can also be derived from
one-loop massive \(N-\) point Feynman diagrams [7]}, [8]}.
In addition Feynman integrals of ladder diagrams with three or four external lines
can be evaluated through the Feynman parametrization and Mellin-Barnes contour
integrals [9]}.
Actually one-loop two-point function \(B_{_0}\)  can be expressed as the linear combination
of the Gauss function \(_2F_1\)  through the recurrence relations respecting the time-space
dimension, one-loop three-point function \(C_{_0}\)  similarly is formulated as the linear
combination of the Appell function \(F_{_1}\) , and one-loop four-point function \(D_{_0}\) 
is formulated as the linear combination of the Lauricella function
\(F_{_s}\)  [10]}, [11]}, respectively.
Those expression for the scalar integral \(C_{_0}\)  is convenient
for analytic continuation and numerical evaluation because continuation of \(F_{_1}\) 
has been analyzed thoroughly. However, how to perform continuation
of the Lauricella function \(F_{_s}\)  outside its convergent domain is still a challenge.
Expressing Feynman integrals as generalized hypergeometric functions in dimension regularization,
the authors of Ref. [12]} analyze the Laurent expansion of these hypergeometric
functions around \(D=4\) . The differential-reduction algorithm to evaluate those hypergeometric
functions can be found in Refs. [13]}, [14]}, [15]}, [16]}, [17]}.
A hypergeometric system of linear PDEs is given through the corresponding Mellin-Barnes
representation [18]}.
Some irreducible master integrals for sunset and bubble Feynman diagrams
are explicitly evaluated through Mellin-Barnes contour in Ref. [19]}.
